Theory of Computing Systems

, Volume 56, Issue 4, pp 630–642 | Cite as

New Lower Bounds on Circuit Size of Multi-output Functions

  • Evgeny Demenkov
  • Alexander S. Kulikov
  • Olga Melanich
  • Ivan Mihajlin


Let B n, m be the set of all Boolean functions from {0, 1} n to {0, 1} m , B n = B n, 1 and U 2 = B 2∖{⊕, ≡}. In this paper, we prove the following two new lower bounds on the circuit size over U 2.
  1. 1.
    A lower bound \(C_{U_{2}}(f) \ge 5n-o(n)\) for a linear function fB n − 1,logn . The lower bound follows from the following more general result: for any matrix A ∈ {0, 1} m × n with n pairwise different non-zero columns and b ∈ {0, 1} m ,
    $$C_{U_{2}}(Ax \oplus b)\ge 5(n-m).$$
  2. 2.
    A lower bound \(C_{U_{2}}(f) \ge 7n-o(n)\) for fB n, n . Again, this is a consequence of the following result: for any fB n satisfying a certain simple property,
    $$C_{U_{2}}(g(f)) \ge \min \{C_{U_{2}}(f|_{x_{i} = a, x_{j} = b}) \colon x_{i} \neq x_{j}, a,b, \in \{0,1\}\} +2n-\varTheta (1)$$
    where g(f) ∈ B n, n is defined as follows: g(f) = (fx 1, … , fx n ) (to get a 7no(n) lower bound it remains to plug in a known function fB n, 1 with \(C_{U_{2}}(f) \ge 5n-o(n)\)).


Circuit complexity Lower bounds Boolean functions Gate elimination 



We would like to thank the anonymous reviewers for their comments that helped us to improve the readability of the paper.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Evgeny Demenkov
    • 1
  • Alexander S. Kulikov
    • 2
  • Olga Melanich
    • 2
  • Ivan Mihajlin
    • 2
  1. 1.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.St. Petersburg Department of Steklov Institute of MathematicsSt. PetersburgRussia

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